A Markov channel is a discrete information channel that includes as special cases the finite state channels and finite state codes of information theory. Kieffer and Rahe proved that one-sided and two-sided Markov channels have the following property: If the input source to a Markov channel is asymptotically mean stationary (AMS), then so is the resulting input-output process and hence the ergodic theorem and the Shannon-McMillan-Breiman theorem hold for the input-output process. Kieffer and Rahe also provided a sufficient condition for any AMS ergodic source to yield an AMS ergodic input-output process. New conditions for a Markov channel to have this ergodicity property are presented and discussed here. Several relations are developed among various classes of channels, including weakly ergodic, indecomposable, and strongly mixing channels. Some connections between Markov channels and the theory of nonhomogeneous Markov chains are also discussed. G IVEN AN INFORMATION SOURCE (a discretetime random process) and a noisy channel (essentially a regular conditional probability measure describing a probability measure on output sequences given an input sequence), information about the source can be communicated to a receiver by first encoding the source sequence into a channel input sequence and decoding the channel output sequence into a reproduction sequence observed by the receiver. Assume that we have some measure of the quality of the reproduction sequence, that is, how well it approximates the original source sequence. The coding theorems of information theory quantify the theoretically optimum performance that can be achieved using the given source and given channel with any encoder and decoder within some constrained class, where "optimum" means that the system has the minimum possible average distortion. The design algorithms of information theory are methods for actually designing codes that work well, ideally not too badly in comparison with the theoretical optimum. The proofs of coding theorems rest primarily on the ergodic theorem and on the Shannon-McMillan-Breiman theorem. They also generally require that the appropriate sample averages converge to constants and Manuscript
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